There is a natural way of going beyond the operations addition, multiplication, exponentiation of ordinals to higher level operations, the next being *tetration* (but not the usual notion of tetration; see below) and *hyper-tetration*, the latter being a term I made up around 1990 but probably not used online until this 4 March 2002 sci.math post. In an analogous way that exponentiation is not suitable for giving explicit formulas to the terms of the transfinite sequence of $\varepsilon$-numbers, ordinal tetration is also not suitable. However, explicit formulas for these terms can be given using ordinal hyper-tetration.

In my answer to the MSE question Why are tetrations not useful? I discussed ordinal tetration a bit. Because that MSE question was closed and thus might not be visible to everyone, I'll repeat the definition and a few relevant comments given there.

**Ordinal Tetration.** Fix an ordinal $\alpha$. We define $\, \sideset{_{}^\beta}{}\alpha \,$ by transfinite induction on $\beta$ as follows.

(*base case*) $\;\; \sideset{_{}^0}{}\alpha = 1\;$ and $\; \sideset{_{}^1}{}\alpha = \alpha$

(*successor case*) $\;\; \sideset{_{}^{\beta + 1}}{}\alpha \; = \; \left(\sideset{_{}^{\beta}}{}\alpha \right)^{\alpha} \;$ for each $\; \beta \geq 1$

(*limit case*) $\;\; \sideset{_{}^{\lambda}}{}\alpha \; = \; \sup \left\{\sideset{_{}^{\beta}}{}{\alpha}: \; \beta < \lambda \right\}\;$ if $\; \lambda \;$ is a nonzero limit ordinal

**Ordinal Tetration vs Usual Tetration.** In the case of finite ordinals (i.e. non-negative integers), this is NOT the same as the usual tetration operation. For instance,
$$\sideset{_{}^4}{}\alpha \; = \; \left( \sideset{_{}^3}{}{\alpha} \right)^{\alpha} \; = \; \left( \left( \sideset{_{}^2}{}{\alpha} \right)^{\alpha} \right)^{\alpha} \; = \; \left( \left( {\alpha}^{\alpha} \right)^{\alpha} \right)^{\alpha} \; = \; {\alpha}^{{\alpha}^{3}} $$
In fact, it can be shown that ("top-down" evaluations are used here)
$$\sideset{_{}^{\epsilon_0}}{}{\epsilon_0} \; = \; {\epsilon_0}^{{\epsilon_0}^{\epsilon_0}} $$
Moreover, we also have $\; \epsilon_0 \; = \; \sup\left\{\omega, \; \sideset{_{}^{\omega}}{}{\omega}, \; \sideset{_{}^{\sideset{_{}^{\omega}}{}{\omega}}}{}{\omega}, \; \ldots \right\}$, and hence it follows that
$$ \epsilon_0 \; = \; {\omega}^{{\omega}^{{\omega}^{{\cdot}^{{\cdot}^{\cdot}}}}} \; = \; \sideset{_{}^{\sideset{_{}^{\sideset{_{}^{\sideset{_{}^{\sideset{_{}^{\cdot}}{}{\cdot}}}{}{\cdot}}}{}{\omega}}}{}{\omega}}}{}{\omega} $$
(In the above, each ellipsis signifies the supremum of all finite-iterated exponentiation/tetration expressions evaluated "top-down".)

**Strong Tetration and Weak Tetration.** There does not seem to be a standard term for this distinction in the literature, probably because this distinction does not arise very often. Mark Neyrinck’s May 1995 undergraduate thesis **An Investigation of Arithmetic Operations** uses the term *top-down* and *bottom-up*. I propose, when these two types of tetration are being discussed together for natural numbers, to use the term *strong tetration* for the ordinary notion of tetration and the term *weak tetration* for the ordinal notion of tetration as defined above.

We now consider the next higher level operation (not mentioned in that MSE answer), which I'll call "hyper-tetration". This is also defined in a "bottom-up" manner.

**Ordinal Hyper-Tetration.** Fix an ordinal $\alpha$. We define $\, \sideset{_{}^{\{\beta\}}}{}\alpha \,$ by transfinite induction on $\beta$ as follows.

(*base case*) $\;\; \sideset{_{}^{\{0\}}}{}\alpha = 1\;$ and $\;\sideset{_{}^{\{1\}}}{}\alpha = \alpha$

(*successor case*) $\;\; \sideset{_{}^{\{{\beta + 1\}}}}{}\alpha \; = \; \sideset{_{}^{\alpha}}{}{\left(\sideset{_{}^{\{\beta\}}}{}\alpha \right)} \;$ for each $\; \beta \geq 1$

(*limit case*) $\;\; \sideset{_{}^{\{\lambda \}}}{}\alpha \; = \; \sup \left\{\sideset{_{}^{\{\beta \}}}{}{\alpha}: \; \beta < \lambda \right\}\;$ if $\lambda$ is a nonzero limit ordinal

The following results are proved in some personal notes of mine that I first wrote around 1990, and which I have revised several times since then:

**Let** $\;\alpha \geq 2.\;$ **The least** $\varepsilon$**-number greater than** $\; \alpha \;$ **is** $\; \sideset{_{}^{\{\omega\}}}{}\alpha.$

**Explicit Expressions for** $\varepsilon$**-Numbers Using Hyper-Tetration:**

**1.** $\;$Let $\;0 \leq n < \omega.\;$ Then $\;\varepsilon_n \; = \;\sideset{_{}^{\{\omega \cdot (n+1)\}}}{}{\omega}.$

**2.** $\;$Let $\;\beta \geq \omega.\;$ Then $\;{\varepsilon}_{\beta} \; = \; \sideset{_{}^{\{\omega \cdot \beta\}}}{}{\omega}.$

These two formulas can be replaced with the single formula $\;{\varepsilon}_{\beta} \; = \; \sideset{_{}^{\{\omega \cdot (1 + \beta)\}}}{}{\omega},\;$ which holds for all ordinals $\beta.$

These results and much more -- formulas for the transfinite sequence of fixed points of $\alpha$'th level operations on ordinal numbers (where $\alpha$ is any finite or transfinite ordinal) -- can be found in *An extended arithmetic of ordinal numbers* by Doner and Tarski (1969; another copy), along with an explanation of why the "bottom-up" formulations for the operations are used (see the first full paragraph on p. 113).